Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130730 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61284 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,23,30,25,29,21,40,24,31,42,48,22,43,41,28)(19,32,35,26,34,45,33,47,20,36,39,46,38,27,37,44);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,24,20)(18,26,31,46)(21,39,41,35)(22,34,25,38)(23,37,42,33)(27,28,45,40)(29,44,43,47)(30,32,48,36);(2,7)(4,13)(6,9)(10,12)(17,20,24,19)(18,27,31,45)(21,37,41,33)(22,34,25,38)(23,39,42,35)(26,28,46,40)(29,32,43,36)(30,44,48,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38295);